FIG. 1 illustrates a conventional direct-down-conversion (DDC) front end receiver 100. At mixers 102a and 102b, an input Radio frequency (RF) signal r(t) is mixed with two orthogonal sinusoids, 2·gI·cos(2π·Fc·t+ΦI) and −2·gQ·sin(2π·Fc·t+ΦK), to generate inphase (Im) and Quadrature (Qm) components respectively, where gI & gQ are amplitudes of sinusoids and ΦI & ΦQ are phases of sinusoids. Ideally, gI=gQ for equality of sinusoid amplitude and Φ1=ΦQ for sinusoid orthogonality. However, and practically, gI≠gQ and ΦI≠ΦQ.
The mixed down baseband (BB) signal (rmbb(t)) generated after mixing is given by rmbb(t)=Im(t)+j·Qm(t), and is represented by the following expression:rmbb(t)=Ksig·sbb(t)+Kimg·sbb*(t)  (1)Where,sbb(t)=desired signal
            K      sig        =                            (                                    1              +                              gⅇ                                  -                  jϕ                                                      2                    )                ⁢                                  ⁢                  K          img                    =              (                              1            -                          gⅇ              jϕ                                2                )                        g      ⁢                          ⁢              (                  relative          ⁢                                          ⁢          gain          ⁢                                          ⁢          imbalance                )              =          gQ      ⁢              /            ⁢      gI                  Φ      ⁢                          ⁢              (                  relative          ⁢                                          ⁢          phase          ⁢                                          ⁢          imbalance                )              =                  Φ        ⁢                                  ⁢        Q            =              Φ        ⁢                                  ⁢        I            
As seen in equation (1), apart from the desired signal (sbb(t))), a scaled version of an undesirable image signal (sbb(t)) also appears due to mixer gain and phase impairments. The magnitude of the image signal depends on the relative phase and gain imbalances.
The Inphase (Im) and Quadrature (Qm) components of the mixed down baseband (BB) signal (rmbb(t)) pass through first and second low pass filters (104a and 104b respectively) to generate filtered Inphase and Quadrature (If) and (Qf) components respectively. The overall baseband signal Rfbb(f) (in frequency domain) after filtering is given by Rfbb(f)=If(f)+j·Qf(f)=Im(f)·HI(f)+j·Qm(f)·HQ(f), and is represented by the following expression:
                                              ⁢                                            Rf              bb                        ⁡                          (              f              )                                =                                                                      K                  sig                                ⁡                                  (                  f                  )                                            ·                                                S                  bb                                ⁡                                  (                  f                  )                                                      +                                                            K                  img                                ⁡                                  (                  f                  )                                            ·                                                S                  bb                  *                                ⁡                                  (                                      -                    f                                    )                                                                    ⁢                                  ⁢                                                      (        2        )                                          Where          ,                                                    K                sig                            ⁡                              (                f                )                                      =                                                            (                                                            1                      +                                                                                                    HQ                            ⁡                                                          (                              f                              )                                                                                                            HI                            ⁡                                                          (                              f                              )                                                                                                      ·                        g                        ·                                                  ⅇ                                                      -                            jϕ                                                                                                                2                                    )                                ·                                  HI                  ⁡                                      (                    f                    )                                                              ⁢                                                                                ⁢                                                                              ⁢              and                                      ⁢                                  ⁢                                            K              img                        ⁡                          (              f              )                                =                                                    (                                                      1                    -                                                                                                                        HQ                            *                                                    ⁡                                                      (                                                          -                              f                                                        )                                                                                                                                HI                            *                                                    ⁡                                                      (                                                          -                              f                                                        )                                                                                              ·                      g                      ·                                              ⅇ                        jϕ                                                                              2                                )                            ·                                                HI                  *                                ⁡                                  (                                      -                    f                                    )                                                      ⁢                                                  =                                          (                                                      1                    -                                                                                            HQ                          ⁡                                                      (                            f                            )                                                                                                    HI                          ⁡                                                      (                            f                            )                                                                                              ·                      g                      ·                                              ⅇ                        jϕ                                                                              2                                )                            ·                              HI                ⁡                                  (                  f                  )                                                                    ⁢                                  ⁢                                  ⁢                              HI            ⁡                          (              f              )                                =                      impulse            ⁢                                                  ⁢            response            ⁢                                                  ⁢            of            ⁢                                                  ⁢            the            ⁢                                                  ⁢            first            ⁢                                                  ⁢            filter            ⁢                                                  ⁢            104            ⁢            a                          ⁢                                  ⁢                                  ⁢                              HQ            ⁡                          (              f              )                                =                      impulse            ⁢                                                  ⁢            response            ⁢                                                  ⁢            of            ⁢                                                  ⁢            the            ⁢                                                  ⁢            second            ⁢                                                  ⁢            filter            ⁢                                                  ⁢            104            ⁢            b                          ⁢                                  ⁢                                  ⁢                              H            ⁡                          (              f              )                                =                                    (                              relative                ⁢                                                                  ⁢                filter                ⁢                                                                  ⁢                imbalance                            )                        =                                          HQ                ⁡                                  (                  f                  )                                            ⁢                              /                            ⁢                              HI                ⁡                                  (                  f                  )                                                                                        (        3        )            
As seen in equation (2), apart from a scaled (Ksig(f)) version of the desired signal (Sbb(f)), a scaled (Kimg(f)) version of the image signal (S*bb(f)) also appears due to the relative filter imbalance. The magnitude of the image signal (S*bb(−f)) depends on the relative filter imbalance H(f)=HQ(f)/HI(f).
The mismatch between the inphase and quadrature components causes image signals, making the use of direct down-conversion unfeasible for multicarrier receivers. It is desirable to calibrate IQ mismatch in baseband receivers with an image rejection ratio (IRR) greater than 90 dB. Further, the IQ mismatch calibration should be done in background, as separate calibration duration cannot be availed. Furthermore, the IQ mismatch calibration time should be less than 500 ms and should not significantly affect the GSM BS boot up time.